Climate Change Adaptation vs. Mitigation:

A Fiscal Perspective

Lint Barrage∗

PRELIMINARY AND INCOMPLETE.

PLEASE DO NOT CITE WITHOUT PERMISSION.

June 17, 2014

Abstract

This study explores the implications of distortionary taxes for the tradeoff between cli-mate change adaptation and mitigation. Public adaptation measures (e.g., seawalls) re-quire government revenues. In contrast, mitigation through carbon taxes raises revenues,but interacts with the welfare costs of other taxes. This paper thus theoretically charac-terizes and empirically quantifies this tradeoff in a dynamic general equilibrium integratedassessment climate-economy model with distortionary Ramsey taxation. First, I find thatpublic investments in adaptive capacity to reduce direct utility impacts of climate change(e.g., biodiversity values) are distorted at the optimum. An intertemporal wedge remainseven when other intertemporal margins are optimally undistorted (i.e., zero capital incometaxes). Second, public adaptation to reduce production impacts of climate change (e.g., inagriculture) should be fully provided to productive effi ciency, even when they are financedthrough distortionary taxes. Third, the central quantitative finding is that the welfare costsof limiting climate policy to adaptation (without a carbon price) may be up to twice ashigh when the distortionary costs of adaptive expenditures are taken into account.

1 Introduction

Adaptation to climate change is increasingly recognized as an essential policy. In the United

States, Federal government agencies have been required to produce climate change adaptation

plans since 2009.1 Governments at all levels are making plans and incurring expenditures for

∗University of Maryland, AREC. Contact email: lbarrage@umd.edu. I thank Werner Antweiler for inspiringme to think about this question, and Yuangdong Qi for excellent research assistance.1 As per Executive Order 13514 (October 5, 2009).

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climate change adaptation,2 such as New York City’s $20 billion plan announced in 2013 in the

aftermath of Hurricane Sandy.3 While a growing academic literature has studied the role of

adaptation in climate policy,4 these studies have generally abstracted from the fiscal implica-

tions of adaptation. In particular, many adaptive measures can only be (effi ciently) provided by

governments (Mendelsohn, 2000). However, when governments raise revenues with distortionary

taxes, interactions between climate and fiscal policy become welfare-relevant. A large literature5

has demonstrated the critical importance of distortionary taxes for the design of pollution miti-

gation policies, such as carbon taxes or emissions trading schemes (see, e.g., review by Bovenberg

and Goulder, 2002). Expanding upon these findings, this paper studies the implications of the

fiscal setting for the optimal policy mix between both climate change mitigation and adaptation.

Specifically, I first theoretically characterize and then empirically quantify optimal adaptation

and mitigation paths in a dynamic general equilibrium integrated assessment climate-economy

model (IAM) with linear distortionary (Ramsey) taxation. More broadly speaking, this paper

thus also builds on recent work on (i) climate policy in macroeconomic models (e.g., Golosov,

Hassler, Krusell, and Tsyvinski, 2014; van der Ploeg and Withagen, 2012; Gerlagh and Liski,

2012; Acemoglu, Aghion, Bursztyn, and Hemous, 2011; Leach, 2009; etc.), (ii) optimal public

goods provision in dynamic Ramsey models (e.g., Economides and Phlippopoous, 2008; Judd,

1999; etc.), (iii) the seminal climate-economy modeling work by Nordhaus (1991; 2000; 2008;

2010; 2013; etc.) as well as the broader IAM literature (e.g., PAGE2009, Hope, 2011; FUND

3.7, Tol and Anthoff, 2013; MERGE, Manne and Richels, 2005; etc.), and, of course, the growing

literature on climate change adaptation versus mitigation.

The increasing policy and academic attention towards climate change adaptation is driven by

three key factors. First, given the current state of international climate policy, substantial warm-

ing is projected by the end of the 21st Century, even with implementation of the Copenhagen

Accord (Nordhaus, 2010).6 Second, due to delays in the climate system, some warming from past

2 For detailed listings of State and local plans in the U.S., see the Georgetown Law School Adaptation Clear-inghouse http://www.georgetownclimate.org/adaptation/state-and-local-plans

3 "Mayor Bloomberg Outlines Ambitious Proposal to Protect City Against the Effects of Climate Change toBuild a Stronger, More Resilient New York" New York City Press Release PR- 201-13, June 11, 2013.

4 Reviewed, e.g., by Agrawala, Bosello, Carraro, Cian, and Lanzi (2011), and discussed in more detail below.5 These include, inter alia: Sandmo (1975); Bovenberg and de Mooij (1994, 1997, 1998); Bovenberg and van

der Ploeg (1994); Ligthart and van der Ploeg (1994); Goulder (1995; 1996; 1998); Bovenberg and Goulder(1996); Jorgenson and Wilcoxedn (1996); Parry, Williams, and Goulder (1999); Goulder, Parry, Williams,and Burtraw (1999); Schwarz and Repetto (2000); Cremer, Gahvari, and Ladoux (2001; 2010); Williams(2002); Babiker, Metcalf, and Reilley (2003); Bernard and Vielle (2003); Bento and Jacobsen (2007); Westand Williams (2007); Carbone and Smith (2008); Fullerton and Kim (2008); Parry and Williams (2010);d’Autume, Schubert, and Withagen (2011); Kaplow (2013); Carbone, Morgenstern, Williams and Burtraw(2013); Barrage (2013); Schmitt (2013); Goulder, Hafstead, and Williams (2014); etc.

6 Specifically, Nordhaus (2010) predicts warming of 3.5C by the year 2100 in the absence of climate policy,and that implementation of the Copenhagen Accord goals will reduce projected warming to only 3.2C ifemissions reductions are limited to wealthy countries.

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emissions will continue even if greenhouse gas emissions stopped today. Third, adaptation plays

a critical role in the fully optimized global climate policy mix (see, e.g., Mendelsohn, 2000). The

literature has studied a variety of questions related to climate change adaptation (for a recent

literature review, see, e.g., Agrawala, Bosello, Carraro, Cian, and Lanzi, 2011). These include

the strategic implications of adaptation in non-cooperative settings (e.g., Antweiler, 2011; Buob

and Stephan, 2011; Farnham and Kennedy, 2010); interactions between adaptation and uncer-

tainty (Felgenhauer and de Bruin, 2009; Shalizi and Lecocq, 2007; Ingham, Ma, and Ulph, 2007;

Kane and Shogren, 2000); comparative statics between macroeconomic variables and optimal

adaptation (e.g., Bréchet, Hritonenko, and Yatsenko, 2013); and the optimal policy mix within

the context of integrated assessment climate-economy models (e.g., Felgenhauer and Webster,

2013; Agrawala, Bosello, Carraro, de Bruin, De Cian, Dellink, and Lanzi, 2010; Bosello, Carraro,

and De Cian, 2010; de Bruin, Dellink, and Tol, 2009; Tol, 2007; Hope, 2006.) There are also

many empirical studies estimating the costs and/or benefits of adaptation in particular settings

and sectors.7 However, to the best of my knowledge, the academic literature has not formally

considered the (differential) implications of the distortionary fiscal policy context for the climate

change adaptation versus mitigation tradeoff.

Of course, concerns about the fiscal impacts of climate change damages and adaptation needs

have been voiced by a number of groups and authors. These include analyses by the U.S. Gov-

ernment Accountability Offi ce (2013), the IMF (2008), and Egenhofer et al. (2010), who provide

a detailed literature review, treatment of key issues, and several case studies focused on the

European Union. Non-governmental organizations such as Ceres have also published reports

emphasizing fiscal costs of climate change (Israel, 2013). Given the policy interest in this topic,

along with the expectation of its importance given the extensive literature on the implications of

distortionary taxes for environmental policy design, this paper thus seeks to contribute to the lit-

erature by formally exploring climate change costs and the optimal adaptation-mitigation policy

mix in a dynamic general equilibrium climate-economy model with distortionary taxation. The

model essentially integrates the COMET climate-economy model with linear distortionary taxes

(Barrage, 2013) with a modified representation of the adaptation possibilities from the AD-DICE

model (2010 version as presented in Agrawala, Bosello, Carraro, de Bruin, De Cian, Dellink, and

Lanzi, 2010). Both the COMET and AD-DICE are IAMs based on the DICE/RICE model-

ing framework developed by Nordhaus (e.g., 2008; 2010). I follow AD-DICE in differentiating

between adaptation capital investments (e.g., sea walls) and flow expenditures (e.g, increased

fertilizer usage). However, as discussed below, my results confirm that I additionally need to

7 For reviews and aggregate studies, see, e.g., IPCC WGII (2007); Parry, Arnell, Berry, Dodman, Frankhauser,Hope, Kovats, and Nicholls (2009); World Bank EACC (2010); and Agrawala, Bosello, Carraro, Cian, andLanzi (2011).

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differentiate between adaptive capacity to reduce climate change impacts on production (e.g, in

agriculture) and direct utility losses (e.g., biodiversity existence values), respectively. The model

thus seeks to accounts for these adaptation types separately.

Before summarzing the results, it should be emphasized that the literature’s estimates of

aggregate adaptation cost functions are at this stage still highly uncertain and require many

strongly simplifying assumptions (Agrawala, Bosello, Carraro, Cian, and Lanzi, 2011). However,

the main research question of this paper is how consideration of the fiscal setting changes the

welfare implications and optimal mix of climate policy for a given (and changeable) adaptation

technology assumed. The three main results are as follows.

First, public funding of flow adaptation inputs to reduce climate damages in the final goods

production sector should remain undistorted regardless of the welfare costs of raising government

revenues. This result is due to the well-known property that optimal tax systems maintain

aggregate production effi ciency under fairly general conditions (Diamond and Mirrlees, 1971).

By noting that public flow adaptation expenditures to reduce production damages are simply a

public input to production, this result also follows directly from studies such as Judd (1999), who

finds that public capital inputs to production should be fully provided even under distortionary

Ramsey taxation.

Second, public funding for both flow and capital adaptation inputs to reduce direct utility

losses from climate change (e.g., biodiversity existence values) should be distorted when govern-

ments have to raise revenues with distortionary taxes. That is, the provision of the climate adap-

tation good should be distorted alongside the consumption of other goods. Perhaps surprisingly,

I find that an intertemporal wedge remains between the marginal rates of transformation and

substitution for adaptation capital investments to reduce utility damages from climate change,

even when it is optimal to have no intertemporal distortions along any other margins (i.e., zero

capital income taxes). That is, adaptation capital investments may be optimally distorted even

when other capital investments should not be.

Third, I find that the welfare costs of relying exclusively on adaptation to address climate

change (i.e., without a carbon price) may be more than twice as large when distortionary tax

instruments are used to raise the necessary revenues. In particular, at a global level, the welfare

costs of relying only on adaptation and of not having a carbon price throughout the 21st Century

are estimated to be $22 trillion in a setting without distortionary taxes, $23-24 trillion when

additional revenue comes from labor or optimized distortionary taxes, and $55 trillion when

capital income taxes are used to raise additional funds ($2005, equivalent variation change in

initial consumption at the global level).

The remainder of this paper proceeds as follows. Section 2 describes the model and derives

the theoretical results. Section 3 discuses the COMET model the calibration of adaptation

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possibilities. Section 4 provides the quantitative results, and Section 5 concludes.

2 Model

The analytic framework extends the dynamic general COMET (Climate Optimization Model of

the Economy and Taxation) model of Barrage (2013) by adding several forms of adaptation. Her

framework builds on the climate-economy models of Golosov, Hassler, Krusell, and Tsyvinski

(2014) and Nordhaus (2008; 2011) by incorporating a classic dynamic optimal Ramsey taxation

framework as presented by Chari and Kehoe (e.g., 1999). Barrage (2013) solves for optimal

greenhouse gas mitigation policies across fiscal scenarios; here we consider adaptation as an

additional choice variable. Specifically, I consider both adaptation capital and flow inputs as

modeled in the 2010 AD-DICE model (Agrawala, Bosello, Carraro, de Bruin, De Cian, Dellink,

and Lanzi, 2010). In addition, I expand upon their framework by separately modeling adaptation

measures that reduce the impacts of climate change on production possibilities and direct utility

damages, respectively.

To briefly preview the model: an infinitely-lived, representative household has preferences

over consumption, leisure, and the environment. In particular, climate change decreases his

utility, but these impacts can be reduced through investments in utility adaptation. There

are two production sectors. An aggregate final consumption-investment good is produced from

capital, labor, and energy inputs. Climate change affects productivity, but the impacts can be

reduced through investments in production adaptation. Carbon emissions stem from a carbon-

based energy input, which is produced from capital and labor. The government must raise a

given amount of revenues as well as funding for climate change adaptation through distortionary

taxes on labor, capital, and carbon emissions.8

The quantitiative model accounts for additional elements such as exogenous land-based emis-

sions, clean energy technology, population growth, and government transfers to households. How-

ever, these features are omitted from the analytic presentation since they do not affect the theo-

retical results. The remainder of this section describes the model in more detail, and derives the

results.

Households

A representative household has preferences over consumption Ct, leisure Lt, and the state of the

climate Tt, along with adaptive capacity to reduce utility damages from climate change Λut (e.g.,

8 In particular, lump-sum taxes are assumed to be infeasible as in the Ramsey tradition. It is moreover assumedthat the revenues raised from Pigouvian carbon taxation are insuffi cient to meet government revenue needs.

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increased land conservation for species preservation). The household takes both the climate and

adaptive capacity Λut as given. That is, adaptation Λu

t is publicly provided.

U0 ≡∞∑t=0

βtU(Ct, Lt, Tt,Λut ) (1)

Pure utility losses from climate change include biodiversity existence value losses, changes

in the amenity value of the climate, disutility from human resettlement, and non-production

aspects of health impacts from climate change (see Barrage, 2013). The benchmark version of

the model assumes additive separability between preferences over consumption, leisure, and the

climate, and that adaptive capacity reduces the disutility from climate change via:

U(Ct, Lt, Tt,Λut ) = v(Ct, Lt) + h[(1− Λu

t )Tt] (2)

Each period, the household allocates his income between consumption, the purchase of one-

period government bonds Bt+1 (at price ρt), and investment in the aggregate private capital

stock Kprt+1. The household’s income derives from net-of-tax labor income wt(1 − τ lt)Lt, net-of-

tax and depreciation capital income 1 + (rt − δ)(1− τ kt)Kprt , government bond repayments

Bt, and profits from the energy production sector Πt. The household’s flow budget constraint

each period is thus given by:9

Ct + ρtBt+1 +Kprt+1 ≤ wt(1− τ lt)Lt + 1 + (rt − δ)(1− τ kt)Kpr

t +Bt + Πt (3)

As usual, the household’s first order conditions imply that savings and labor supply are

governed by decision rules:

UctUct+1

= β 1 + (rt+1 − δ)(1− τ kt+1) (4)

−UltUct

= wt(1− τ lt) (5)

where Uit denotes the partial derivative of utility with respect to argument i at time t.

Production

The final consumption-investment good is produced with a constant returns to scale technology

using capital K1t, labor L1t, and energy Et inputs, assumed to satisfy the standard Inada condi-9 As in Barrage (2013), I assume that (i) capital holdings cannot be negative, (ii) consumer debt is bounded

by some finite constantM via Bt+1 ≥ −M , (iii) purchases of government debt are bounded above and belowby finite constants, and (iv) initial asset holdings B0 are given.

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tions. In addition, output is affected by both the state of the climate Tt and adaptive capacity

in final goods production, Λyt :

Yt = F1t(L1t, K1t, Et, Tt,Λyt ) (6)

= [1−D(Tt)(1− Λyt )] · A1tF1t(L1t, K1t, Et)

where A1t denotes an exogenous total factor productivity parameter. The modeling of climate

change prodution impacts in this way was pioneered by Nordhaus (e.g., 1991). Production

impacts include productivity losses in sectors such as agriculture, fisheries, and forestry, changes

in labor productivity due to health impacts, impacts of changes in ambient air temperatures on

energy inputs required to produce a given amount of heating or cooling services, etc. (see, e.g.,

Nordhaus, 2007; Nordhaus and Boyer, 2000).

Profit maximization and perfect competition in final goods production implies that marginal

products of factor inputs, denoted by F1it for input i at time t, are equated to their prices in

equilibrium:

F1lt = wt (7)

F1Et = pEt

F1kt = rt

Carbon-based energy inputs are assumed to be producible from capital K2t and labor L2t

inputs through a constant returns to scale technology:

Et = A2tF2t(K2t, L2t) (8)

With perfect competition and constant returns to scale, profits from energy production Πt

will be zero in equilibrium:

Πt = (pEt − τEt)Et − wtL2t − rtK2t (9)

where pEt represents the price of energy inputs and τEt is the carbon tax.10

The numerical COMET model further considers an emissions reduction technology wherein a

fraction of emissions µt can be abated at a cost Ωt(µt), as in the DICE model (Nordhaus, 2008).

For ease of illustration, and since it does not affect the analytic results, this section abstracts

from a representation of this technology.

10 Energy inputs are represented in terms of tons of carbon-equivalent; one unit of energy thus equals one tonof carbon emissions.

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Both capital and labor are assumed to be perfectly mobile across sectors. Profit maximization

thus implies that prices and marginal factors will be equated via:

(pEt − τEt)F2lt = wt (10)

(pEt − τEt)F2kt = rt

Government: Fiscal and Climate Policy

The government faces two tasks: raising revenues to meet an exogenous sequence of expenditure

requirements Gt > 0∞t=0 and choosing an optimal policy mix to address climate change. Fol-

lowing recent work in the adaptation-mitigation literature (e.g., Felgenhauer and Webster, 2013;

Agrawala et al., 2010; de Bruin, 2011), I model adaptive capacity in sector i, Λti, as an aggregate

of both adaptation capital KΛit (e.g., seawalls) and flow adaptation inputs λit (e.g., additional

fertilizer):

Λit = f i(KΛi

t , λit) (11)

Each period, the government thus needs revenues to finance government consumption Gt,

the repayment of bonds Bt, flow adaptation expenditures λyt and λ

ut , and net new investment

in adaptation capital stocks KΛ,it . The government receives revenues from the issuance of new

one-period bonds Bt+1, by levying linear taxes on labor and capital income, and through carbon

taxes. The government’s flow budget constraint is thus given by:

Gt +Bt + λyt + λut +KΛ,yt+1 +KΛ,u

t+1 = τ ltwtLt + τEtEt + τ kt(rt − δ)(KΛ,yt +KΛ,u

t ) + ρtBt+1 (12)

The specification (12) differentiates itself from the standard Ramsey setup as in Chari and

Kehoe (1998) through the inclusion of carbon taxes and adaptation expenditures, and differs

from the climate-economy Ramsey model in Barrage (2013) through adaptation expenditures.

Finally, given (12), we can summarize the market clearing conditions for the different capital

stocks in the economy at time t :

Kt = K1t +K2t +KΛ,yt +KΛ,u

t (13)

= Kprt +KΛ,y

t +KΛ,ut (14)

Where private capital is composed of final good and energy production sector capital: Kprt ≡

K1t + K2t. Specification (13) assumes that, over the 10-year period considered in the model,

capital is perfectly mobile across sectors. I moreover impose that depreciation rates are identical

across sectors, although this assumption can easily be relaxed. Finally, I assume throughout that

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the government can commit to a sequence of tax rates at time zero.

Climate System

The quantitative COMET model uses the 2010-DICE representation of the climate system and

carbon cycle. However, for the purposes of this analytic section, the only assumption made is

that temperature change Tt at time t is a function Ft of initial carbon concentrations S0 and all

past carbon emissions:

Tt = zt (S0, E0, E1, ..., Et) (15)

where:∂Tt+j∂Et

≥ 0 ∀j, t ≥ 0

Competitive Equilibrium

A Competitive equilibrium ("CE") in this economy can now be defined as follows:

Definition 1 A competitive equilibrium consists of an allocation Ct, L1t, L2t, K1t+1, K2t+1, Et, Tt, λyt , λ

ut , K

Λ,y, KΛ,ut ,

a set of prices rt, wt, pEt, ρt and a set of policies τ kt, τ lt, τEt, BGt+1 such that

(i) the allocations solve the consumer’s and the firm’s problems given prices and policies,

(ii) the government budget constraint is satisfied in every period,

(iii) temperature change satisfies the carbon cycle constraint in every period, and

(iii) markets clear.

The social planner’s problem in this economy is to maximize the representative agent’s lifetime

utility (1) subject to the constraints of (i) feasibility and (ii) the optimizing behavior of households

and firms. I will follow the primal approach (see, e.g., Chari and Kehoe, 1999), which solves for

optimal allocations after having shown that and how one can construct prices and policies such

that this optimal allocation will be decentralized by optimizing households and firms. The

optimal alloation - the Ramsey equilibrium - is formally defined as follows:

Definition 2 A Ramsey equilibrium is the CE with the highest household lifetime utility for a

given initial bond holdings B0, initial aggregate private capital Kpr0 and abatement capital KΛ,y

0

and KΛ,y0 , initial capital income tax τ k0, and initial carbon concentrations S0.

Following the standard approach, one can now set up the primal planner’s problem as per

the following proposition:

Proposition 3 The allocations Ct, L1t, L2t, K1t+1, K2t+1, Et, Tt, λyt , λ

ut , K

Λ,y, KΛ,ut , along with

initial bond holdings B0, initial aggregate private capital Kpr0 and abatement capital KΛ,y

0 and

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KΛ,y0 , initial capital income tax τ k0, and initial carbon concentrations S0 in a competitive equi-

librium satisfy:

Yt + (1− δ)Kt + λyt + λut ≤ Ct +Gt +Kt+1 (RC)

Tt ≥ zt(S0, E0, E1, ...Et)] (CCC)

Et ≤ F2t(AEt, K2t, L2t) (ERC)

L1t + L2t ≤ L2t (LC)

K1t +K2t +KΛ,y +KΛ,ut ≤ Kt (KC)

and∞∑t=0

βt [UctCt + UltLt] = Uc0 [Kpr0 1 + (Fk0 − δ)(1− τ k0)+B0] (IMP)

In addition, given an allocation that satisfies (RC)-(IMP), one can construct prices, debt holdings,

and policies such that those allocations constitute a competitive equilibrium.

Proof: See Appendix. In words, Proposition 1 implies that any allocation satisfying the six

conditions (RC)-(IMP) can be decentralized as a competitive equilibrium through some set of

prices and policies. Throughout the remainder of this paper, I assume that the solution to the

Ramsey problem is interior and that the planner’s first order conditions are both necessary and

suffi cient. The planner’s problem is thus to maximize (1) subject to (RC)-(IMP) (see Appendix).

Results

Before discussing the results, one more definition is required. The marginal cost of public funds

(MC /F ) is a measure of the welfare cost of raising an additional dollar of government revenues.

When governments can use lump-sum taxes to raise revenues, then the marginal cost of public

funds is equal to 1, as households give up $1 in a pure transfer. However, when revenues are

raised through distortionary taxes, the costs of raising $1 will equal $1 plus the excess burden

(or marginal deadweight loss) of taxation. Barrage (2013) presents a GDP-weighted estimate

of the MCF based on a review of the literature estimating the MCF across countries and tax

instruments equal to 1.5, implying that on average $0.50 is lost for every $1 of government

revenue raised. Following the literature on optimal pollution taxes and distortionary taxes,

formally define the marginal cost of public funds in this model as equal to the ratio of public to

private marginal utility of consumption:

MCFt ≡λ1t

Uct(16)

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The wedge between the marginal utility of public and private income thus serves as a measure

of the distortionary costs of the tax system.

Given (16), we can state the theoretical results. First, as formally demonstrated in the

Appendix, the optimality conditions for the optimal public provision of flow adaptation inputs

to guard against production damages is given by:

(−F1TtD(Tt)) =1

f yλt(17)

where F1Tt denotes the marginal production losses in the final output sector due to a change

in temperature at time t, D(Tt) is the damage function, and fyλtindicates the marginal change in

total adaptive capacity due to an increase in the flow adaptation input for production damages.

Intuitively, the left-hand side of equation (17) thus measures the marginal benefit of increasing

adaptive capacity in final goods production Λyt : marginal damages F1Tt will be reduced by

a fraction of D(Tt).11 The right-hand side indicates the marginal cost of increasing adaptive

capacity Λyt through an increase in flow adaptation expenditures λ

yt . While expression (17) will

be evaluated at different allocations depending on the tax system of the economy, we thus see

that, as such the expression does not depend on the marginal cost of public funds. That is,

the optimal provision of flow adaptive expenditures to reduce damages in production is thus not

distorted, even when other margins in the economy, such as labor supply, are distorted. I first

discuss this and three other results somewhat informally, and then proceed to summarize the

theoretical findings in a formal proposition.

Result 1 Public funding of flow adaptation inputs to reduce climate damages in the final goodsproduction sector should remain undistorted regardless of the welfare costs of raising gov-

ernment revenues. That is, flow adaptation to reduce production damages should be fully

provided in the Ramsey equilibrium.

As discussed further below, Result 1 is a consequence of the well-known property that optimal

tax systems maintain aggregate production effi ciency under fairly general conditions (Diamond

and Mirrlees, 1971). Similarly, by noting that public adaptation expenditures to reduce pro-

duction damages are simply a public input to production, the logic of Result 1 follows directly

from studies such as Judd (1999), who explores optimal provision of public capital inputs to

production under distortionary Ramsey taxation. Judd (1999) finds that public flow productive

inputs should always be fully provided, regardless of the distortionary costs of raising revenues.

Next, and in contrast, consider the optimality condition governing the provision of flow adap-

tation expenditures to reduce utiltiy impacts of climate change (derived in the Appendix):11 This is because the derivative of net damage term ΛytD(Tt) with respect to adaptive capacity Λyt equals

D(Tt). See equation (6).

11

(−UTtTt)Uct︸ ︷︷ ︸MRS

1

MCFt︸ ︷︷ ︸wedge

=1

fuλt︸︷︷︸MRT

(18)

The first term on the left-hand side of equation (18) is the representative household’s marginal

rate of substitution (MRS) between the final consumption good and adaptive capacity to reduce

climate change utility impacts. The right-hand side equals the marginal cost of increasing this

adaptive capacity, or the marginal rate of transformation (MRT) between the final consumption

good and adaptive capacity (through increased flow expenditures λut ). However, contrary to

equation (17) here there is a wedge between the MRS and MRT equal to the marginal cost

of public funds. This wedge is proportional to the distortionary cost of the tax system - the

marginal cost of public funds.

Result 2 Public funding of flow adaptation inputs to reduce direct utility losses from climate

damages is distorted when governments have to raise revenues with distortionary taxes.

That is, the provision and thus consumption of the climate adaptation good should be dis-

torted alongside the consumption of other goods when governments must impose distor-

tionary taxes.

The wedge between the MRT and MRS for the flow adaptation good for utility damages in

(18) can intuitively be thougt of as an implicit tax on the consumption of the climate adaptaiton

good. Just like any consumption good in this economy, the climate adaptation good will be

’taxed’by being less-than-fully provided (compared to a setting with lump-sum taxation).

Importantly, it should be noted that both Result 1 and Result 2 are analogous extensions

of findings in the environmental tax interaction literature regarding the internalization of en-

viornmental impacts affecting utility versus production (Bovenberg and van der Ploeg, 1994;

Williams, 2002; Barrage, 2013). These studies all find that the optimal pollution tax formula

internalizes production damages ’fully’(without a wedge), whereas a large literature has demon-

strated that utility damages are ’less-than-fully’internalized (with a wedge) when there are other,

distortionary taxes (see, e.g., review by Bovenberg and Goulder, 2002).

Next, I consider optimal public investment in adaptation capital, which are governed by the

following optimality conditions (derived in the Appendix):

λ1t

βλ1t+1

= (1− δ) +[D(Tt+1)Yt+1f

yKt+1

](19)

λ1t

βλ1t+1

= (1− δ) +

[(−UTt+1Tt+1)

Uct+1

fuKt+1

]1

MCFt+1

(20)

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In contrast, the government’s optimality condition for private capital investment’s is given

by:

λ1t

βλ1t+1

= (1− δ) + [F1kt+1] (21)

Intuitively, conditions (19)-(21) indicate that the marginal social return on investments in

each of the different types of capital be equated at the optimum. In order to interpret these

conditions further, it is helpful to consider the wedges (or lack thereof) between intertemporal

marginal rates of substitution and transformation implied by (19)-(21).

First, there are multiple marginal rates of transformation between consumption in periods t

and t+ 1. On the one hand, Ct can be transformed into Ct+1 through investments in the private

final goods production capital, K1t+1 :

MRTK1tct,ct+1 =

−1

Fkt+1 + (1− δ) (22)

However, Ct can also be transformed into Ct+1 through investments in climate change adap-

tation in the production sector. In particular, the return to giving up −1 units of Ct to invest

in KΛ,yt yields a reduction in time t + 1 climate damages of f yKt+1 percent, or f

yKt+1 ·D(Tt+1)Yt

units of the consumption-investment good:

MRTKΛyct,ct+1 =

−1

f yKt+1D(Tt+1)Yt + (1− δ)(23)

Comparison of (22)-(23) with 19) and (21) immediately yields the standard condition that

the marginal rate of transformation between Ct and Kt+1 is equated across private and public

adaptation capital. In addition, comparison of (21) with the household’s Euler Equation (4)

demonstrates the well-known result that the optimum features no intertemporal wedge wheneverλ1t

βλ1t+1= Uct

βUct+1(see, e.g., Chari and Kehoe, 1999). That is, if the planner’s intertemporal MRS

equals the household’s MRS, then the optimal allocation features no wedge between the MRT

and MRS of consumption across time period after period t > 0.

Result 3 The optimal policy in period t > 0 features undistorted (full) public investment in

adaptation capital to reduce production damages from climate change if and only if the

optimal policy leaves private capital investment undistorted (zero capital income tax). In

this case, the government should invest fully in production adaptation capital even though

the necessary revenues are raised with distortionary taxes.

Finally, and in contrast, consider investment in adaptation to utility damages. For this type

of capital, the relevant intertemporal margin is between consumption today Ct and utility from

13

the climate amenity tomorrow, −Tt+1. Giving up −1 units of Ct to marginally increase utility

adaptation capital KΛ,ut+1 decreases climate change impacts on utility by f

uKt+1, and thus increases

the amount of the climate amenity in utility by −Tt+1fuKt+1 units. In addition, this investment

will leave (1−δ)KΛ,ut+1 units of the final consumption-investment good available after depreciation.

Denominated in equivalent units of the climate amenity, the value of an increase in KΛ,ut+1 by one

unit is thus iven by (1−δ)Uct+1−UTt+1 . In sum, the MRT between Ct and the consumption good Tt+1

based on investments in adaptation capital is thus given by:

MRTKΛuCt,T t+1 =

−1

−Tt+1fuKt+1 + (1−δ)Uct+1−UTt+1

(24)

The representative agent’s MRS between Ct and Tt+1 as a consumption good is conversely

given by:

MRSCt,Tt+1 =−βUTt+1

Uct(25)

In order for investments in utility damages adaptation capital to be undistorted (i.e., no

intertemporal wedge), it must thus be the case that:

MRSCt,Tt+1 = MRTKΛuCt,T t+1

⇔Uct

βUct+1

= (1− δ) +(−UTt+1Tt+1)

Uct+1

fuKt+1 (26)

Comparison between (??) and (20) immediately leads to the final theoretical result:

Result 4 The optimal policy at time t > 0 leaves investment in adaptation capital to reduce

direct utility impacts from climate change distorted if governments raise revenues through

distortionary taxes (specifically if MCFt+1 > 1). In addition, optimal investment in utility

adaptation capital remains distorted even if it is optimal for there to be no distortions on

investment in private capital (no capital income tax) or public adaptation capital to reduce

production impacts of climate change.

When the marginal cost of raising public funds in the future exceeds unity, it is easy to see

from comparison beween (??) and (20) that the optimal allocation features a wedge between

the intertemporal MRS and MRT of the consumption good today Ct and the climate amenity

tomorrow, −Tt+1. That is, when MCFt+1 > 1, we find that MRSCt,Tt+1 6= MRTKΛuCt,T t+1. Perhaps

surprisingly, this wedge remains even if the necessary condition for no intertemporal wedge in

private and public production adaptation holds(

λ1tβλ1t+1

= UctβUct+1

).

14

The theoretical results discussed so far can be formally summarized by the following propo-

sition.

Corollary 4 If preferences are of either commonly used constant elasticity form,

U(Ct, Lt) =C1−σt

1− σ + ϑ(Lt) + v(Tt(1− Λut )) (27)

U(Ct, Lt) =(CtL

−γt )1−σ

1− σ + v(Tt(1− Λut )) (28)

then, after period t > 1 :

(i) investment in private capital should be undistorted (the optimal capital income tax is zero),

(ii) investment in public adaptation capital to reduce climate change production damages

should be undistorted;

(iii) investment in public adaptation capital to reduce climate change direct utility damages

in period t should be distorted in proportion to the marginal cost of raising public funds in period

t+ 1;

(iv) public flow adaptation expenditures to reduce climate change production damages should

be undistorted (satisfy productive effi ciency);

(v) public flow adaptation expenditures to reduce climate change direct utility damages in

period t should be distorted in proportion to the marginal cost of raising public funds in period t;

and:

(ii) the optimal carbon tax is implicitly defined by:

τ ∗Et = τPigou,YEt +τPigou,UEt

MCFt(29)

Proof: See Appendix. Intuitively, the proof follows straightforwardly from Results 1 − 4

described above along with the observation that preferences of the form (27) or (28) imply thatλ1t+1λt

= Uct+1Uct

for all t. It should be noted that the optimality of zero capital income taxes in

periods t > 1 for these preferences is the classic Chamley-Judd result (Chamley, 1986; Judd,

1985) as subsequently demonstrated, e.g., by Chari and Kehoe (1999).

The expression implicitly defining the optimal carbon tax (29) is identical in form to the one

presented by Barrage (2013) for this model without adaptation.12 Introducing non-degenerate

government revenue needs (to finance adaptation) does thus not lead to a change in the for-

mulation defining the optimal carbon tax. Intuitively, this is the case because carbon taxes in

this model fall on energy inputs, which are an intermediate good. Consequently, carbon is not a

12 Of course, the value of the optimal tax will differ between the models as the implicit expression is evaluatedat different allocations when there are adaptation possibilities.

15

desirable tax base in excess of the internalization of the externality from climate change damages

(see, e.g., Diamond and Mirrlees, 1971; also discussion by Goulder, 1996). Consequently, use-

ful additional revenues to finance adaptation will be raised through other means, such as labor

income taxes.

The theoretical results presented in this Section are limited to qualitative statements based

on implicit expressions. In order to solve the model and assess the quantitative importance of the

fiscal context for the optimal adaptation-mitigation policy mix and the welfare costs of climate

change, the next Section describes the numerical implementation and calibration of the model.

3 Adaptation Cost Function Calibration

Several special challenges arise in the calibration of adaptation cost functions. On the one hand,

bottom-up studies are limited both in terms of sectors and regions covered. In addition, bottom-

up studies often do not report their results in suffi ciently comparable metrics that would permit

straightforward integration into a single cost function. Finally, estimating adaptation costs in

certain sectors is extraordinarily diffi cult. For example, with regards to ecosystems and species

preservation, the best estimate identified by the UNFCCC (2007) was based on a study that

estimated the costs of increasing the amount of gloablly protected lands. Whether and to which

extent such a policy would reduce climate change impacts on ecosystems is extremely diffi cult to

quantify. Consequently, existing aggregate adaptation cost functions are often acknowledged to

be highly uncertain, and require many simplifying assumptions (see discussion by, e.g., Agrawala,

Bosello, Carraro, Cian, and Lanzi, 2011).

A number of IAM studies on adaptation rely on cost functions backed out from the DICE/RICE

model family (Nordhaus, 2011; Nordhaus and Boyer, 2000; Nordhaus, 2008, etc.) The DICE

damage function uses adaptation-inclusive cost estimates in sectors such as agriculture. Studies

such as de Bruin, Dellink, and Tol (2009) thus seek to split the DICE damage function into

gross damages and adaptation costs, calibrating the model such that the benchmark result du-

plicates the DICE net damages path. As discussed by Agrawala, Bosello, Carraro, Cian, and

Lanzi (2011), additional studies based on this kind of approach include Bahn et al. (2010), Hof

et al. (2009), and Bosello et al. (2010). Other studies and models, notably the FUND model

(Tol, 2007) features sector-specific adaptation to sea-level rise, based on bottom-up adaptation

cost studies of specific measures such as building dikes. Finally, the PAGE model (Hope et

al., 1993; Hope, 2006, 2011) features (exogenous) adaptation variables which can reduce climate

damages in several distinct ways. The PAGE model also differentiates between adaptation to

economic and non-economic impacts of climate change. This study uses a modified version of the

adaptation cost and gross damage estimatesunderlying the calibration AD-DICE/-RICE model,

16

as detailed in Agrawala, Bosello, Carraro, de Bruin, de Cian, Dellink, and Lanzi (2010). The

authors combine a backing-out prodedure based on the DICE/RICE models with results from

other adaptation studies (e.g., Tan and Shibasaki (2003) on agriculture) and modelers’ judg-

ment to provide adaptation cost and effectiveness estimates across the sectors and regions of the

RICE model. The key modifications required to use their estimates in this paper is to sepa-

rately estimate adaptation and gross damage functions for production and utility damages, and

to recalibrate so as to reproduce benchmark results in the context of the COMET model.

I focus on public adaptation efforts. In reality, climate change adaptation consists of both

pulic and private actions, as discussed by Mendelsohn (2000). I thus exclude adaptation to

climate change impacts on the value of leisure time use, as those are unambigously private

(see AD-DICE 2010).13 In the other sectors, adaptation will likely be a mix of public and

private actions.14 However, private adaptation costs that are borne by (competitive) firms are

equivalent to public adaptation costs in the COMET model, as both figure analogously into the

economy’s resource constraint of the final consumption-investment good. In other words, since

the government is assumed to have to raise a given amount of revenue regardless of climate

change adaptation, decreasing aggregate output by one unit affects the problem equivalently

to an increase in required government expenditure by one unit. The modeling of adaptation

costs in the COMET is thus only with a loss of generality if those costs are actually borne by

households. To the extent that the latter case would lead to non-separability between preferences

over climate change, consumption, and leisure, this would be expected to change the results, as

prior research on optimal emissions taxes and non-separability has shown (see, e.g., Schwarz and

Repetto, 2002; Carbone and Smith, 2008). While this is an important area for future research,

in the current setting I focus on adaptation costs borne by the public and production sectors as

those are presumably much larger in magnitude than household-level adaptation costs.

4 Calibration

In order to assess the quantitative importance of the distortions discussed above, I integrate an

explicit adaptation choice into the Climate Optimization Model of the Economy and Taxation

(COMET) presented by Barrage (2013). The COMET is based on the seminal DICE climate-

economy modeling framework (Nordhaus, 2008, 2010, etc.). It is a global growth model with two

production sectors: a final consumption-investment good is produced using capital, labor, and

13 That is, for time use impacts, I exclude the estimated adaptation costs and retain net damages in the costand damages aggregation based on AD-DICE 2010.

14 For example, IFPRI (2009) estimates the costs of offsetting climate change impacts on nutrition throughagricultural research, rural roads, and irrigation. Research and roads - presumably both public goods - areestimated to account for close to 60% of optimal adaptation efforts in 2050.

17

energy inputs, and energy is produced from capital and labor. Production further depends on

the state of the global climate. There is both clean and carbon-based energy. Consumption of

the latter leads to carbon emissions which accumulate in the atmosphere and change the climate.

The climate system is modeled exactly as in DICE, with three reservoirs (lower ocean, upper

ocean/biosphere, atmosphere) and including exogenous land-based emissions. The COMET

differs from DICE in several key ways necessary to incorporate a simple representation of fiscal

policy. First, households have preferences over consumption, leisure, and the climate (seperably).

A globally representative government faces the dual task of raising revenues and addressing

climate change. The government can issue bonds and impose linear taxes on labor, capital income,

and energy inputs. It faces an exogenous sequence of government consumption requirements and

household transfer obligations. These are calibrated based on IMFGovernment Finance Statistics

to match globally representative government spending patterns. See Barrage (2013) for details.

Importantly, adaptation to climate change is only implicitly considered in the COMET

through the damage function (based on DICE), which is net of adaptation. This study thus

extends the COMET by adding gross damage functions and adaptation choice variables to re-

duce climate change impacts on both production processes and utility. In order to maintain

comparability to the literature and as a benchmark, I calibrate the COMET gross damage and

adaptation cost functions based on regional-sectoral estimates underlying the AD-DICE model

and as presented by Agrawala, Bosello, Carraro, de Bruin, de Cian, Dellink, and Lanzi (2010).

The AD-DICE model aggregates these estimates into a single gross damage and adaptation cost

function; however in the setting with distortionary taxes, both damages and adaptation needs to

be considered separately, as demonstrated in the theoretical section above. In order to provide

separate estimates for production and utility damages, I thus disaggregate the AD-DICE esti-

mates according to the same procedure as outlined in Barrage (2013). Specifically, the different

sectoral damages are disaggregated according to:

18

Impact/ Adaptation Category Classification

Agriculture Production

Other vulnerable markets (energy Production

services, forestry production, etc.)

Sea-level rise coastal impacts Production

Amenity value Utility

Ecosystems Utility

Human (re)settlement Utility

Catastrophic damages Mixed

Health Mixed

Table 1: Climate Damage Categorization

Health impacts are classified as affecting both production and utility as losses in available time

due to disease reduces both work time endowments and leisure time. The COMET consequently

converts disease-adjusted years of life lost into an equivalent TFP loss from a reduction in the

global aggregate labor time endowment, and values the non-work share of time losses at standard

value of statistical life figures (see Barrage (2013) for details). I follow the same approach here

to convert the gross (of adaptation) damage estimates presented by Agrawala et al. (2010) into

gross production and utility losses from health impacts of climate change.

Catastrophic impacts of climate change are also assumed to affect both production possibili-

ties and utility directly. I assume that the relative importance of production/utility impacts of a

severe climate event in each region is proportional to the relative importance of production/utility

impacts across the other sectors outlined in Table 1.15 I weight both damages and adaptation

costs in each region by the predicted output based on the 2010-RICE model (Nordhaus, 2011) in

the relevant calibration year. and re-aggregating across regions and sectors leads to the following

results for climate change impacts and optimal adaptation at 2.5C :

15 However, as in Barrage (2013), time use values are excluded from the calculation of the distribution ofcatastrophic impacts across production/utility damages, as such severe events are assumed to affect predom-inantly the other impact sectors.

19

Moment Target Model Target Source:

Opt. max temperature change (C) 2.96 2.99 COMET without dist. taxes and without adaptation

Opt. Y−adaptation effect at ∼ 2.5C 52% 44% Re-aggregation of AD-DICE Damages

Opt. U−adaptation effect at ∼ 2.5C 60% 47% Re-aggregation of AD-DICE Damages

Opt. Y-adaptation cost at ∼ 2.5C (% GDP) 0.47% 0.32% Re-aggregation of AD-DICE Damages

Opt. U-adaptation cost at ∼ 2.5C (% GDP) 0.16% 0.13% Re-aggregation of AD-DICE Damages

Opt. Carbon Tax ($/mtC in 2015) 70 68 COMET without dist. taxes and without adaptation

Table 2: COMET Adaptation Calibration Targets and Results

COMET Adaptation Calibration Estimates at 2.5C

Production (Y ) Utility (U) AD-DICE (Agg.)

Gross Damages (% GDP) 2.2% 0.7% 2.25%

Flow Adaptation Cost (% GDP) 0.24% 0.03% 0.17%

Stock Adaptation Cost (% GDP) 0.23% 0.12% 0.21%

Total Adaptation Cost (% GDP) 0.47% 0.15% 0.38%

Adaptation Effectiveness (% Damages Avoided) 52% 60% 0.46%

Residual Damages 1.1% 0.3% 1.21%

The modified damage and adaptation estimates are thus generally higher than the aggregated

results in AD-DICE. Such difference may arise, for example, due to alternative weights used in the

global aggregation of damages, which in the COMET are based on predicted 2065 global output

shares (that weight developing regions’impacts relatively more than 2010 weights). Additional

challenges in directly adoping AD-DICE parameters arise from the generally different structure

of the COMET which features endogenous investment, labor supply, energy production, etc.

Table 2 summarizes the key moments I seek to match in the calibration of gross damages and

adaptation costs, as well as the actual model output for the benchmark COMET model run

without distortionary taxes. That is, I seek to match the following moments in the COMET

before introducing distortionary taxes (specifically while allowing lump-sum taxation).

The functional forms used to model gross damages and adaptive capacity are chosen and

modified relative to AD-DICE as follows. The AD-DICE model considers gross damages (as a

fraction of GDP) given by GD = α1Tt + α2Tα3t . I maintain their functional form to represent

both production and utility damages, but adjust the damage function by parameters θy and θu in

order to match benchmark estimates of gross damages at 2.5C as discussed above. Specifically,

I thus assume that net output is given by:

Yt =

(1

1 + (1− Λyt ) · θy[α1Tt + α2T

α3t ]

)· AtF1(K1t, L1t, Et) (30)

20

where Λyt represents adaptive capacity (fraction of damages reduced) as in the theoretical

model outlined above. Similarly, utility is modeled as having preferences over an environmental

good that diminishes with climate change according to:

U =(Ct(1− φlt)γ)1−σ

1− σ +1

1− σ

(1

1 + (1− Λut ) · θu [α1 + α2T

α3t ]

)1−σ

(31)

Finally, I use the AD-DICE functional form used to represent the adaptation technology:

Λit = β1

(β2(λit)

ρ + (1− β2)(KΛ,it )ρ

)β3ρ

(32)

In order to match the desired moments indicated above, I have to adjust several of the

parameters in (30)-(32) relative to what is used in the AD-DICE model. With these adjustments,

however, the benchmark (no distortionary taxes) COMET results arguably replicate the desired

moments reasonably well, as shown in Table 2.

5 Quantitative Results

The analysis focuses on four central fiscal scenarios:

1. A "First-Best" scenario without distortionary taxes, where the government can raise rev-

enues through lump-sum taxation. This is the benchmark scenario used to calibrate the

model to match the literature, as discussed in the preceding section. The literature typically

assumes no distortionary taxes.

2. A "Total Tax Reform" scenario with fully optimized distortionary taxes.

3. A "Green Tax Reform, τ k Revenue Recycle" scenario where labor income tax rates are held

fixed at baseline levels (35.19%). Carbon tax revenues can be used ("recycled") to reduce

capital income taxes and/or to finance adaptaiton expenditures.

4. A "Green Tax Reform, τ l Revenue Recycle" scenario where capital income tax rates are

held fixed at baseline levels (39.35%). Carbon tax revenues can be used to reduce labor

income taxes and/or to finance adaptaiton expenditures.

Figure 5 shows the evolution of optimal temperature change16 across these four scenarios.

While the differences across scenarios are small, I find that the optimal amount of climate change

tolerated across scenarios is increasing in the welfare costs of distortionary taxes.

16 Throughout this paper, temperature change refers to mean atmospheric surface temperature change overpre-industrial levels in degrees Celsius.

21

2000 2050 2100 2150 2200 22501

1.5

2

2.5

3

3.5

Year

Opt

imal

Tem

pera

ture

Cha

nge

(Deg

rees

Cel

sius

)

Optimal Temperature Change Across Fiscal Scenarios

First­Best (Lump­Sum Taxes)Total Tax Reform (Optimized Dist. Taxes)Green Tax Reform (BAU Taxes,τk Rev. Recycle)

Green Tax Reform (BAU Taxes,τL Rev. Recycle)

Intuitively, the reason for this finding is that carbon taxes exacerbate the welfare costs of

other taxes. This finding is in line with the longstanding previous literature on this topic (see,

.e.g., Goulder, 1996; Bovenberg and Goulder, 2002, etc.). Indeed, I find that optimal climate

change mitigation is lower, the more distortionary the tax system, as demonstrated in Figure 5 :

The corresponding carbon tax schedules are depicted in Figure 5. Again, the key result is

that optimal carbon levies are generally lower when there are other, distortionary taxes, in line

with the previous literature.17

Finally, Figures 5 and 5 show the amount of adaptation to climate change impacts on pro-

duction and utility, respectively. In particular, the graphs depict the fraction of climate change

impacts avoided due to both flow and capital investments in adaptive capacity.

The results suggest that the government engages in more adaptation in the fiscal scenarios

where the tax code is more distortionary. This result may seem counterintuitive in light of

the theoretical results, which demonstrated that both flow and capital investments in utility

adaptation should be distorted in proportion to the marginal cost of public funds. However, it is

17 It should be noted that the carbon tax is defined as the difference between total taxes imposed on carbon-based and clean energy. This is because, in the Green Tax Reform with τk Revenue Recycling, the governmentimposes a tax on all types of energy in addition to the carbon tax. The reason for this tax is that both typesof energy usage increase the tightness with which the fixed labor income tax constraint binds.

22

2000 2020 2040 2060 2080 2100 2120 2140 2160 21800

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Year

Miti

gatio

n (%

Cle

an E

nerg

y)

Optimal Mitigaton (Clean Energy Share) Across Fiscal Scenarios

First­Best (Lump­Sum Taxes)Total Tax Reform (Optimized Dist. Taxes)Green Tax Reform (τk Rev. Recycle)

Green Tax Reform (τL Rev. Recycle)

2000 2020 2040 2060 2080 2100 21200

100

200

300

400

500

600

700

Year

Car

bon

Tax

($/m

tC)

Optimal Carbon Tax Paths Across Fiscal Scenarios

First­Best (Lump­Sum Taxes)Total Tax Reform (Optimized Dist. Taxes)Green Tax Reform (BAU Taxes,τk Rev. Recycle)

Green Tax Reform (BAU Taxes,τL Rev. Recycle)

23

2000 2020 2040 2060 2080 2100 2120 2140 2160 21800

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Year

Out

put A

dapt

atio

n (%

Dam

age

Red

uctio

n)

Optimal Adaptation to Output Losses Across Fiscal Scenarios

First­Best (Lump­Sum Taxes)Total Tax Reform (Optimized Dist. Taxes)Green Tax Reform (BAU Taxes,τk Rev. Recycle)

Green Tax Reform (BAU Taxes,τL Rev. Recycle)

2000 2020 2040 2060 2080 2100 2120 2140 2160 21800

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Year

Util

ity A

dapt

atio

n (%

Dam

age

Red

uctio

n)

Optimal Adaptation to Utility Losses Across Fiscal Scenarios

First­Best (Lump­Sum Taxes)Total Tax Reform (Optimized Dist. Taxes)Green Tax Reform (BAU Taxes,τk Rev. Recycle)

Green Tax Reform (BAU Taxes,τL Rev. Recycle)

24

important to remember that these distortions are only relative to the relevant marginal rates of

transformation within a given fiscal scenario. In general equilibrium, these values change across

scenarios. Indeed, the theoretical results demonstrated that investments in adaptation to reduce

output losses from climate change should maintain productive effi ciency regardless of the tax

system. Consequently, the differences in optimal adaptation across fiscal scenarios for output

losses depicted in Figure 5 are due to general equilibrium differences across the fiscal scenarios,

rather than due to wedges affecting the government’s adaptation decisions.

Many of the current climate change policy efforts in the United States (and other countries)

focus on adaptation rather than mitigation. As a final quantitative exercise, I thus compute the

global welfare costs of failing to engage in mitigation over the 21st Century, and of pursuing

an adaptation-only policy. For each fiscal scenario, these calculations compare welfare with

optimized carbon taxes against a scenario where the planner cannot enact carbon prices until

2115. Welfare is measured as equivalent variation change in initial period aggregate consumption

(C2015 in $2005). Table 3 provides the results.

Policy Scenario: ∆Welfare1

Income Taxes: Carbon Tax: $2005 bil.

First-Best None (until 2115) -

First-Best Optimized 21, 663

Optimized None (until 2115) -

Optimized Optimized 23, 596

BAU+ RR(τ k)2 None (until 2115) -

BAU+ RR(τ k)2 Optimized 54, 634

BAU+ RR(τ l)2 None (until 2115) -

BAU+ RR(τ l)2 Optimized 22, 7341Equivalent variation change in agg. initial consumption ∆C2015

Table 3: Welfare Costs of Adaptation-Only Policy

The results presented in Table 3 suggest that the welfare costs of addressing climate change

only through adaptation may be more than twice as large when the revenues to finance adaptation

measures are raised through distortionary taxes. The welfare costs of failure to engage in carbon

taxes over the 21st Century are estimated to be $22 trillion ($2005 present value equivalent

variation) in the benchmark setting with lump-sum taxation. When adaptation measures have to

be financed through either labor taxes or an optimized tax mix, this cost increases to around $23-

24 trillion. However, perhaps most importantly, when adaptation is paid for through increased

capital income taxes, the welfare costs of an adaptation-only climate policy increases to $55

25

trillion. While the optimal amount of adaptation pursued is thus slightly higher when there

are distortionary taxes, the welfare costs of pursuing this policy may be considerably larger.

While there are tremendous quantitative uncertainties surrounding the estimates presented in

this section, this finding arguably presents at least a notable warning that the public financing

of climate change adaptation expenditures warrants further attention.

6 Conclusion

Adaptation to climate change impacts is increasingly recognized as a critical public policy issue.

Even countries without major national mitigation policies, such as the United States, are working

towards integrated adaptation policies, as exemplified by President Obama’s newly created Task

Force on Climate Preparedness and Resilience. A growing academic literature has explored the

policy tradeoffs between adaptation and mitigation from a variety of perspectives, such as based

on strategic implications (e.g., Antweiler, 2011; Buob and Stephan, 2011).

This study revisits this question from a fiscal perspective. In particular, when governments

raise revenues through distortionary taxes, the fiscal costs of climate policy become welfare-

relevant. On the one hand, public financing for adaptive capacity requires government revenues.

On the other hand, mitigation policies such as carbon taxes raise revenues, but are also well-

known to exacerbate the welfare costs of other taxes (see, e.g., Goulder, 1996; Bovenberg and

Goulder, 2002). I therefore use a dynamic general equilibrium climate-economy model with linear

distortionary taxes to theoretically characterize and empirically quantify these tradeoffs.

First, I find that both flow and capital expenditures towards adaptation to reduce direct

utility impacts of climate change (e.g., biodiversity existence value losses) are distorted at the

optimum. Furthermore, an intertemporal wedge between the marginal rates of transformation

and substitution for adaptation capital investments to reduce utility damages from climate change

remains even when other intertemporal margins are optimally left undistorted (e.g., zero capital

income tax).

Second, adaptation to reduce climate change impact on final goods production should be fully

provided to maintain productive effi ciency, regardless of the welfare costs of raising government

revenues. This result follows directly from studies such as Judd (1999), and is based on the

well-known property of Ramsey tax systems that they maintain aggregate production effi ciency

onder mild conditions (Diamond and Mirrlees, 1971).

Third, the quantitative analysis suggests that the welfare costs of relying exclusively on adap-

tation to address climate change (i.e., without a carbon tax) may be up to twice as large in a

setting with distortionary taxes. The benchmark model runs suggest a global welfare costs of

an adaptation-only policy throughout the 21st Century to be $22 trillion in a setting without

26

distortionary taxes, $23-24 trillion when additional revenue comes from labor or optimized dis-

tortionary taxes, and $55 trillion when capital income taxes are used to raise additional funds

($2005, equivalent variation change in initial consumption at the global level). While these figures

are based on highly uncertain adaptation cost estimates, they nonetheless show that the fiscal

setting warrants further attention in considering the tradeoff between climate change adaptation

and mitigation.

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32

7 Appendix

7.1 Proof of Proposition 1

This proof follows closely the one presneted by Barrage (2013), but adds the adaptation variables

of this study. In addition, let Ωt denote public transfers to households. These are not in the

analytic model above but are featured in the quantitative COMET model and thus incorporated

in the poof here.

Before proceding, it is useful to write out the firm’s and household’s first order conditions.

Given the appropriate convexity assumptions, I can assume that the solution to the problem is

interior. Let γt denote the Lagrange multiplier on the consumer’s flow budget constraint (3) in

period t, his first order conditions are given by:

[Ct] :

γt = βtUct (33)

[Lt] :−UltUct

= wt(1− τ lt) (34)

[Kprt+1] :

γt = βγt+1 1 + (rt+1 − δ)(1− τ kt+1) (35)

[Bt+1] :

Uctρt = βUct+1 (36)

The climate variable Tt and utility adaptation Λut do not enter his problem directly because

he takes both values as given, and because of the additive separability in preferences we have

assumed in (2).

Next, the final goods producer’s problem is to select L1t, K1t, and Et to solve:

maxF1t(L1t, K1t, Et, Tt,Λyt )− wtL1t − pEtEt − rtK1t

where the firm takes the climate Tt and public adaptation Λyt as given.

Defining Fjt as the first derivative of the production function with respect to input j, the

firm’s FOCs are:

F1lt = wt (37)

F1Et = pEt

F1kt = rt

33

The energy producer’s problem is to maximize:

max(pEt − τEt)Et − wtL2t − rtK2t

subject to:

Et = F2t(L2t, K2t)

With FOCs:

(pEt − τEt)F2lt = wt (38)

(pEt − τEt)F2kt = rt

Proof Part 1: If the allocations and initial conditions constitute a competitive equi-librium, then the constraints (RC)-(IMP) are satisfied. In a competitive equilibrium,

the consumer’s FOCs (33)-(36) must be satisfied. Multiplying both sides of (35) by Kt gives:

[γt − γt+1 1 + (rt+1 − δ)(1− τ kt+1)

]Kprt+1 = 0 (39)

Equivalently, for bond holdings, we find:

[γtρt − γt+1

]Bt+1 = 0 (40)

Next, consumer optimization dictates that the transversality conditions must hold in a com-

petitive equilibrium:

limt→∞

γtBt+1 = 0 (41)

limt→∞

γtKprt+1 = 0

Lastly, the consumer’s flow budget constraint (3) holds in competitive equilibrium. Multiply-

ing both sides of the flow budget constraint in each period by the Lagrange multiplier γt leads

to:

γt[Ct + ρtBt+1 +Kpr

t+1

]= γt [wt(1− τ lt)Lt + 1 + (rt − δ)(1− τ kt)Kpr

t +Bt + Ωt + Πt] (42)

As discussed above, the assumptions of perfect competition and constant returns to scale in

34

the energy sector imply that equilibirum profits will be equal to zero.18 Summing equation (42)

over all t thus yields:

∞∑t=0

γt[Ct + ρtBt+1 +Kpr

t+1 − wt(1− τ lt)Lt − 1 + (rt − δ)(1− τ kt)Kprt −Bt − Ωt

]= 0 (43)

All terms relating to capital and bond holdings after period zero cancel out of equation (43)

out as can be seen by substituting in from (39), (40) and the transversality conditions (41). We

thus end up with:

∞∑t=0

γt [Ct − wt(1− τ lt)Lt − Ωt] = γ0 [Kpr0 1 + (r0 − δ)(1− τ k0)+B0] (44)

Finally, one can substitute out for the remaining prices γt, wt(1 − τ lt), and r0 in (44) from

the consumer’s and firm’s FOCs in order to obtain the implementability constraint (IMP):

∞∑t=0

βt [UctCt + UltLt − UctΩt] = Uc0 [Kpr0 1 + (Fk0 − δ)(1− τ k0)+B0] (45)

We have demonstrated that the implementability constraint is satisfied in a competitive

equilibrium.

The last step is where adaptation comes into play directly. We need to show that the final

goods resource constraint (RC) holds in competitive equilibrium. Start by adding up the con-

sumer and government flow budget constraints (3) and (12) with the addition of transfers to

households Ωt in each. Canceling redundant terms on each side leaves:

Gt + λyt + λut +Kabtt+1 + Ct +Kpr

t+1 = wtLt + τEtEt + Πt + (1− δ + rt)Kprt + (1− δ)Kabt

t

Next, invoking the definition of energy sector profits, substituting in based on the labor and

capital market clearing conditions, and substituting in for factor prices based on the energy

producer’s FOCs (38) changes the RHS to:

18 One can formally confirm this by substituting the energy producer’s FOCs for labor and capital inputs intothe definition of energy sector profits:

Πt = (pEt − τEt)F (K2t, L2t)− Fl2t(pEt − τEt)L2t − Fkt(pEt − τEt)K2t

If F (K2t, L2t) exhibits constant returns to scale, then by Euler’s theorem for homogenous functions,(F (K2t, L2t) = Fl2tL2t + Fk2tK2t), and the profits expression reduces to zero.

35

Gt + λyt + λut +Kabtt+1 + Ct +Kpr

t+1 = wtL1t + pEtEt + rtK1t + (1− δ)Kprt + (1− δ)Kabt

t (46)

By Euler’s theorem for homogenous functions, (46) becomes the resource constraint, as de-

sired:

Gt + λyt + λut +Kabtt+1 + Ct +Kpr

t+1 = Yt + (1− δ)Kprt + (1− δ)Kabt

t (47)

Finally, the carbon cycle constraint (CCC) and the energy producer’s resource constraint

(ERC) hold by definition in competitive equilibrium.

Direction: If constraints(RC)-(IMP) are satisfied, one can construct competitiveequilibrium. I proceed with a proof by construction. First, set factor prices as equal to their

marginal products evaluated at the optimal allocation:

F1lt = wt (48)

F1Et = pEt

F1kt = rt

These factor prices are clearly consistent with profit maximization in the final goods sector,

as needed in a competitive equilibrium. Next, set the return on bonds based on the consumer’s

intertemporal first order conditions for bond holdings (36):

ρt = βUct+1/Uct

Again, this price is obviously consistent with utility maximization. Proceed similarly in

setting the labor income tax bsaed on the household’s labor supply and consumption FOCs:

−Ult/Uct = (1− τ lt)Flt

1 +Ult/UctFlt

= τ lt

Next, let the tax rate on capital income for each time t > 0 be defined by the household’s Euler

equation and the firm’s capital holdings FOC:

Uct = βUct+1 1 + (F1kt+1 − δ)(1− τ kt+1)

τ kt+1 = 1− Uct/βUct+1 − 1

(F1kt+1 − δ)

36

As before, being defined by the consumer and firm’s FOCs these tax rates will clearly be

consistent with utility and profit maximization.

Proceeding in the same manner, define the carbon tax based on the energy and final goods

producers’FOCs (38) and (37) as:

τEt = pEt −F1lt

F2lt

Finally, in order to construct bond holdings in period t, multiply the consumer budget con-

straint (3) by its Lagrange multiplier γt and sum over all periods from period t onwards:

∞∑s=t

γs[Cs + ρsBs+1 +Kpr

s+1 − ws(1− τ ls)Ls − 1 + (rs − δ)(1− τ ks)Kprs −Bs − Πs − Ωt

]= 0

(49)

The consumer’s FOCs and transversality conditions (41) must necssarily hold in a competitive

equilibrium, indicating that all future terms relating to capital and bond holdings in (49) cancel

out. We are thus left with:

∞∑s=t

γs [Cs − ws(1− τ ls)Ls − Πs − Ωt] + γt 1 + (rt − δ)(1− τ kt)Kprt = γtBt (50)

Use the agent’s and the firms’FOCs once again to substitute out prices in equation (50)

finally leads to:19

∞∑s=t

βs−tUcsUct

[Cs +

UlsUcs

Ls − Ωs

]+Uct−1

βUctKprt = Bt

Given allocations, this equation defines the unique bond holdings that are consistent with a

competitive equilibrium.

Since the prices and policies defined as outlined above are all based on household and firm

optimality conditions, they are clearly consistent with utility and profit maximization. It thus

remains to be shown that the constraints necessary for competitive equilibrium are satisfied as

well. First, the final goods resource constraint, the carbon cycle constraint, the energy produc-

tion resource constraints, and the factor market clearing conditions for labor and different capital

19 For the capital return in period t, note that the substitution derives from:

γt [rkt(1− τkt) + (1− δ)]Kprt

= βtUct

[Uct−1βUct

]Kprt

= βt−1Uct−1Kprt

37

types all hold by assumption. If we can show that the consumer budget constraint is satisfied,

then by Walras’ law it follows that the government budget constraint must be satisfied also.

Following the standard line of reasoning (see, e.g., Chari and Kehoe, 1999), we can note the fol-

lowing. First, only the consumer’s competitive equilibrium-budget constraint is relevant to show

that our constructed prices, bond holdings, and policies are constitute a competitive equilibrium.

In a competitive equilibrium, the household’s intertemporal budget constraint must hold, along

with the consumer’s FOCs and the consumer’s transversality conditions. Consequently, (39) and

(40) must hold in a competitive equilibrium as well . However, as shown in the first part of this

proof, at the prices selected above, the consumer’s competitive equilibrium-budget constraint

is identical to the implementability constraint, which holds by assumption. The competitive

equilibrium budget constraint at the chosen prices is thus satisfied, as was to be shown .

7.2 Proof of Theory Results

The social panner’s problem is given by:

maxk

∞∑t=0

βt[[v(Ct, Lt) + h[(1− Λut )Tt] + φ [UctCt + UltLt]]︸ ︷︷ ︸≡Wt

+∞∑t=0

βtλ1t

[ [1−D(Tt)(1− Λy

t )] · A1tF1t(L1t, Et, K1t)

+ (1− δ)Kt

−Ct −Kt+1 −Gt − λyt − λut

]

+∞∑t=0

βtξt[Tt −zt(S0, E0, E1, ...Et)] (51)

+∞∑t=0

βtλlt [Lt − L1t − L2t]

+

∞∑t=0

βtλkt

[Kt −K1t −K2t −KΛ,y −KΛ,u

t

]+∞∑t=0

βtωt [F2t(AEt, K2t, L2t)− Et]

+∞∑t=0

βtηyt

[f y(KΛ,y

t , λyt )− Λyt

]+∞∑t=0

βtηut

[fu(KΛ,u

t , λut )− Λut

]−φ Uc0 [K0 1 + (Fk0 − δ)(1− τ k0)]

38

The associated first-order conditions for periods t > 0 are as follows:

[Et] :

λ1tFEt −∞∑t=0

ξt+j∂Tt+j∂Et

= ωt (52)

[Tt] :

UTt + λ1tF1Tt = ξt (53)

[Lt] :

Ult = −λlt

[L2t] :

λlt = ωtF2lt (54)

[L1t] :

λ1tF1lt = λlt (55)

Conditions (52)-(55) can be used to derive an expression implicitly defining the optimal

carbon tax. Specifically, combining (54) and (55) thus yields:

λ1tF1lt

F2lt

= ωt (56)

Combining (56) with the FOCs for energy inputs (52) and temperature change Tt (53), we

obtain:

FEt −∞∑j=0

βj[UTt+jλ1t

+λ1t+j

λ1t

F1Tt+j

]∂Tt+j∂Et

=F1lt

F2lt

(57)

Comparison between (57) and the energy firm’s optimality condition (10) at equilibrium

factor prices (which are equated with marginal products), it thus immediately follows that the

optimal carbon tax is implicitly defined by:

τ ?Et =

∞∑j=0

βj[UTt+jλ1t

+λ1t+j

λ1t

F1Tt+j

]∂Tt+j∂Et

(58)

In words, expression (58) is the present value sum of all future marginal utility and production

damages associated with an additional ton of carbon emissions in period t. This expression is

analogous to the one derived in Barrage (2013); however it will be evaluated at a different

allocation due to the introduction of adaptation possibilities.

Next, consider the following additional FOCs for the planner’s problem, again for t > 0 :

[Λut ] :

− UTtTt = ηut (59)

39

[λut ] :

λ1t = ηutfuλt (60)

[Λyt ] :

λ1tD(Tt)Yt = ηyt (61)

where Yt denotes gross output (before climate damages).

[λyt ] :

λ1t = ηytfyλt

(62)

Combining (59)-(60) yields the optimality condition for public provision of flow adaptation

inputs λut :

−UTtTtλ1t

=1

fuλt(63)

(−UTtTt)/UctMCFt

=1

fuλt(64)

Multiplying the left-hand side of (63) by Uct/Uct and invoking the definiton of the MCF in

(16) thus yields the desired result of equation (18).

Similarly, combining (61)-(62) yields the optimality condition for flow adaptation inputs for

production damages λyt :

D(Tt)Yt =1

f yλt(65)

Finally, consider the planner’s FOCs relating to optimal adaptation capital for t > 0 :

[KΛ,yt ] :

λkt = ηytfyKt

[KΛ,ut ] :

λkt = ηutfuKt

[Kt+1] :

λ1t = βλ1t+1(1− δ) + βλkt+1

[K1t] :

λ1tF1kt = λkt

Based on these equations, we can derive intertemporal optimality conditions for the economy’s

capital stocks. First, for production adaptation, we have that:

λ1t = βλ1t+1(1− δ) + β[ηyt+1f

yKt+1

](66)

40

Substituting in based on the shadow value of production adaptation (61), equation (66)

becomes:

λ1t

βλ1t+1

= (1− δ) +[D(Tt+1)Yt+1f

yKt+1

](67)

Similarly, for utility adaptation capital, we obtain:

λ1t = βλ1t+1(1− δ) + β[ηut+1f

uKt+1

]Again substituting in based on the shadow value of utility adaptation from (60) leads to:

λ1t = βλ1t+1(1− δ) + β[(−UTt+1Tt+1) fuKt+1

]= βλ1t+1(1− δ) + λ1t+1β

[(−UTt+1Tt+1)

λ1t+1

fuKt+1

]And hence, invoking again the definition of the MCF in (16) leads to the desired result:

λ1t

βλ1t+1

= (1− δ) +

[(−UTt+1Tt+1)

Uct+1

1

MCFt+1

fuKt+1

](68)

41